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Doel: your assertion that Adolfo Barabino was wanting to demonstrate that ET was inferior simply proves the point that it is, for there was no such conscious intention at all. The fact that it may have an unconscious effect is another matter - the temperament moves the sensitive pianist . . . so if there is any difference between the quality of performance, it's the temperament giving the effect, not the pianist - let alone the assertion that he thought about it. The ET concert was in a particularly prestigious venue at a castle in the middle of a lake and there was not a shimmering of the conscious intention you ascribe to his performance.

Jake - I'm with Ed Foote on this. The point is that ET of any sort whatever cannot give the effects that we hear in these performances of cycling through keys and then finding the lock to which the one marked "home" fits . . .

In this concept, ET is akin to a torture of perpetual sex without finding orgasm. Romantic music cannot cope with that. ET literally makes it impotent.

There is another simile which I believe to be valid relating to the mathematics of music and in particular Entropy*. Much of music has within it contrasts of order vs disorder and God is to be found within "Order" (Pythagorean Music of the Spheres) equating with "God". This was particularly part of the consciousness of composers who were Freemasons. (I realise that I'm leaning towards the link about Ernest McClain and apologise as I have not read this thread in detail so apologise for any duplication of ideas which might have sprung to life before). The effect of this, however, is that the music cycles through periods of turmoil seeking ways of resolution and the finding of order. With ET, the sound is always disordered and order cannot be found.

It's for this reason that UT enured into 19th century romanticism and composers who were conscious of what they were doing would find chords, even in disordered keys, that locked into order. This is found in Morski's performance http://www.youtube.com/watch?v=zNFRTrO8XA4 and is apparent in Miena Senada's playing of the 4th Ballade under the direction of Adolfo Barabino where the temperament pointed to the finding of order amidst the turmoil on the 4th Ballade http://www.youtube.com/watch?v=fJT5Q6HooyA at 10:09 - and the visitor comment to that video is interesting. However, that tuning was before I had started looking at the TuneLab IH facility to assist the harmonic locking process.

(The Chopin 2nd sonata 3rd movement is such a piece where the first chord locks whilst the second is disordered.)

_________________________
_______________________________David Pinnegar, B.Sc., A.R.C.S.- East Grinstead, Sussex, UK - - http://www.organmatters.com - _______________________________Restoring life to music . . . and music to life . . . and a good deal more!

On the theme of keys and finding the one that fits the lock to home, one can perceive the music as travelling on a succession of journies, and the contrast between travelling and being still. Even more, finding "Home".

Many of the current world seek how to be still but cannot escape from always being moving. In ET one cannot find that stillness.

I hope that these recordings and these ideas reached independently of Ed Foote but clearly in parallel perhaps bring his concepts to life and understanding perhaps from an easier to understand perspective and backed up with a solid aural demonstration.

Aldfredo - I am with you on your efforts to bring harmonic accordance to the instrument to make the sound sound as one rather than a collection of notes - as it is exactly that that I have been trying to achieve using the locking together of the Unequal Temperament to lock together the natural intervals of the harmonics as far as might be achievable and as distorted by the inharmonicity inherent of piano strings.

My intervention in this thread is hopefully to demonstrate to you the use of your techniques in tuning unequal temperaments in the same way. Whilst it is a start, piano tuning is not as simple as getting out the Korg instrument tuner and tuning to the needle, as the piano does not behave like that - and that's the interest we find in these discussions . . .

Best wishes

David P

Edited by Unequally tempered (11/01/1109:47 PM)

_________________________
_______________________________David Pinnegar, B.Sc., A.R.C.S.- East Grinstead, Sussex, UK - - http://www.organmatters.com - _______________________________Restoring life to music . . . and music to life . . . and a good deal more!

ED wrote:â€¦"expansion of the 12 semitones is stretching, which has nothing to do with equality."â€¦

Perhaps it hasn't been acknowledged that 12 root of two ET is a geometric progression, it is the first attempt at gaining a "tuning form" that should be extended beyond the temperament octave, a precise geometry that should include the entire range of notes. In fact, every semitone is strung together by the scale incremental ratio.

As a consequence it should be evident that, in order to talk about ET theory and normal tuning, the expansion of the 12 semitones is fundamental. Understanding how deeply "expansion" had to do with our historical ET and has to do with any "equal" form is first step. "Expansion" may explain why tuners could never put 12 root of two into practice. And eventually it will explain why ET needed to string together also our octaves, that is what Chas ET has accomplished.

So David, for the time being let's leave orgasms aside and let me ask you too the same basic question: How do you tune ET thirds, fourths, fifths, octaves, 12ths and 15ths? How do you manage the expansion of 12 semitones and the tuning of 88 notes into that notorious, precise, geometric form? What are the rules of your normal ET tuning?

So David, for the time being let's leave orgasms aside and let me ask you too the same basic question: How do you tune ET thirds, fourths, fifths, octaves, 12ths and 15ths? How do you manage the expansion of 12 semitones and the tuning of 88 notes into that notorious, precise, geometric form? What are the rules of your normal ET tuning?

Dear Alfredo

I entered this thread not to upset you but simply because my work and results support what I saw Ed Foote saying on this thread and are therefore relevant, and add further in the directions he has indicated.

How do I tune ET thirds?

I don't.

I don't tune ET.

How do I tune ET fifths and 12ths?

I don't: I don't tune ET.

This afternoon I was working with Kazimierz Morski on a recording and he was playing Mussorgsky and Ravel without at all needing the services of ET. ET is an outmoded 20th century encumbrance which has dulled the senses.

In the days when a 55 comma scale was used and semitones were recognised as major and minor semitones http://www.hoboy.net/Hoboy/BeyondTemperament.html"Good musicians know how to use the different effects of the intervals, and prove their value by the expresÂ­sion and variety they are able to draw from them."

Equal temperament is a system of blunting the differences between keys, robbing them of their eccentricities and colours, for people drilled mechanically into shape as machines in corporations . . . . to behave like robots. And leading ultimately to "music" constructed in boys bedrooms on computers to be played by machine at nightclubs engendering jumping together as machines and subsequent behaviour as robots.

Of course the above opinion is extreme, and I don't believe it . . . entirely . . . but put it in such terms merely to make the point.

I tune many fifths pure, many twelths pure - and with string inharmonicity leading to sharp harmonics, the bass notes go a bit flatter. Where the 17th accords with the intervals more pure than ET in the UT temperament, often I strike a balance between the 12th and the 17th, often with the 15th being spot on . . . and in this way, the temperament enables the tuning to hold together as a resonant whole. Sometimes I even listen to the 7th harmonic, the flat 21st, and make sure that it's not screaming against something. Music with thirds comes out sweet in keys that it's meant to, and shimmer where it's meant to, and music with 5ths comes out solid - an effect which Chopin exploited with 5ths in one hand and shimmering confusion in the other . . .

Morski comes to the piano as a conductor to an orchestra, and this too is Barabino's approach. The number of classical symphonies which feature ending on a chord with a prominent major third is significant. Standard ET on the piano ruins them.

Were you to apply your mathematical approach of achieving better harmonic structure as you have with ET but to the UT temperaments you would do the musical world a considerable service.

Best wishes,

David P

Edited by Unequally tempered (11/02/1102:28 PM)

_________________________
_______________________________David Pinnegar, B.Sc., A.R.C.S.- East Grinstead, Sussex, UK - - http://www.organmatters.com - _______________________________Restoring life to music . . . and music to life . . . and a good deal more!

I think that, tuning-wise, the closer to ET a modern piano is (i.e. the best compromise between progressive 5th and M3rds) then the better it sounds for any period of music.

Chris - have you been following my YouTube recordings? ET leads to a reliable and constant shimmering in all keys. Is that what "real" music is about? Would you rather live in a temporary inflatable house with wobbling walls and floor or prefer one with at least a solid floor for your feet to be on the ground?

:-)

Best wishes

David P

_________________________
_______________________________David Pinnegar, B.Sc., A.R.C.S.- East Grinstead, Sussex, UK - - http://www.organmatters.com - _______________________________Restoring life to music . . . and music to life . . . and a good deal more!

Were you to apply your mathematical approach of achieving better harmonic structure as you have with ET but to the UT temperaments you would do the musical world a considerable service.

Best wishes,

David P

This is a lovely idea--if Alfredo were to, as an experiment, build a UT.

By the way, David, do you know when Adolfo Barabino plans to release his album in the UT that you are using?

(I must say that I hope the sound is similar to that on the videos. I actually prefer it to the sound of the albums, or at least to the sound of the albums as they are reproduced on his site--the sound on the cd's seems to have more compression applied than I like, while the sound on your videos, while not perfect, is more natural and intimate--the sound of a piano when we are sitting near the piano. Is he recording it at Hammerwood, using your pianos there, by any chance?)

..."Would you rather live in a temporary inflatable house with wobbling walls and floor or prefer one with at least a solid floor for your feet to be on the ground?"...

That would depend on what comes along with your idea of "solid floor". You don't seem to be talking about one reliable architecture nor a smooth geometry. Have you checked the laying of your (floor) tiles? Then, why "temporary"? Why "inflatable"? Why "wobbling" as the only possible alternative? And why "a...house" to live in, instead of THE house to live over?

The old CDs were done by a professional company before I met him and before he discovered the Hammerwood UT Bechstein . . .

Finding a recording company that is interested is proving to be a bit of a challenge but we are working on it.

The videos' sound are done with a H2 Zoom recorder about a meter away from the instrument_______| ___)|_/ *

and aren't processed in any way - so they are as natural as can be . . .

However, after 27 years of concerts the hammers are worn and need refelting so are about to go away to Abel. Today I was working with Kazimierz Morski who wanted to record the instrument and, whether or not it is recent concerts (not Adolfo's special touch) that have been the straw to break the camel's back, but under the microscope of the microphone played through the detail of Lowther speakers, unevenness of the hammers began to show up and was actually quite annoying. So a good recording of the instrument will have to be on hold.

Alfredo - I was trying to get across in my description of ET the situation where everything is moving. Without purity anywhere other than octaves, and with IH octaves aren't "pure" either, nothing is still, nothing is focussed. In contrast in UT, keys cause music to shift and then be still, and also to "lock".

Kazimierz Morski's partner is a Professor of Aesthetics and Philosophy in Music in Italy and has written on the subject "What is Music?" so it's no coincidence that Kazimierz wanted to play and to record the instrument at Hammerwood in UT . . .

I'm not writing wanting to annoy you but simply to encourage your mathematics to UTs as well as ET.

Best wishes

David P

Edited by Unequally tempered (11/03/1104:39 PM)

_________________________
_______________________________David Pinnegar, B.Sc., A.R.C.S.- East Grinstead, Sussex, UK - - http://www.organmatters.com - _______________________________Restoring life to music . . . and music to life . . . and a good deal more!

Chris - have you been following my YouTube recordings? ET leads to a reliable and constant shimmering in all keys. Is that what "real" music is about? Would you rather live in a temporary inflatable house with wobbling walls and floor or prefer one with at least a solid floor for your feet to be on the ground?

Yes, David, I have seen your videos and followed this argument for some time now in this forum so I don't want to flame another war. You create a false impression that key colour is a universal concept in musical appreciation throughout musical history, and throughout musical forms, rather than an artefact of keyboard design. For example, no vocal or chamber group would ever have any desire to deliberately produce a temperament-derived key colouration. However, they would be influenced if a keyboard is part of the group.

On the other hand, and grant it, the vibrato produced by harpsichords with their upper harmonics under ET is a bit sour, so there is a natural reason to temper the tuning and then be restricted to certain optimal keys. Then there was the concept of relaxation and tension derived from contrasting chord purity.

Modern pianos are a different beast and temperament-derived vibrato sounds much more acceptable. The development of modern pianos therefore fostered and permitted the development of harmonic and tonal complexity not possible during earlier times . Key colour became less important than the developing richness of the harmonic language. ET was a natural requirement of this harmonic complexity.

For your analogy, I would rather have a house that wobbles consistently all over than a house that is half steady but where you liable to fall through unforeseen rotten floor sections.

Thanks for your reply - and I agree with your argument that there are inconsistencies but . . .

I'm tuning modern Steinways as well as older instruments and on one the owner's mother comments (dispassionately and as a lay listener familiar with the instrument's standard sound) that she's never heard it so "in tune". The tweaking of temperament in the remote keys is not actually a great deal away from ET but the result is most wonderful purity elsewhere. When harmonic accordance is achieved, to which Alfredo is striving, then the sound in the badly tuned keys becomes more pure but is musically interesting.

Furthermore, now with considerable experience of the repertoire and its relationship with UT, there are many composers of the 19th century who drove phrasing and musical construction through passages of turmoil in order to resolve them by way of stillness and purity. The chords chosen, even in poorly tuned UT keys chosen for "special effect" avoided notes of poor intonation. We have noted passages where Liszt quite litterally wanted to achieve the effect of skating lightly as if upon glass or ice, particularly to find solid land on the other side of the phrase or passage. Ubiquitous ET has blinded us to these effects.

It may well be that the _beauty_ achieved by UT recordings does not come through in recording and standard electronic reproduction, particularly through the poor medium of computer compressed audio. But in real life, in the room, the effect can be particularly moving and, close to the piano, where effects are heard which cannot be recorded http://www.youtube.com/watch?v=0JpSH4YTypE

So in real life, rather than the mere abstraction of recorded sound particularly as by YouTube http://www.youtube.com/watch?v=nPvHq8HvTKg the performer is particularly aware, and possibly even more so than the audience, of the particular effects and this influences phrasing and accentuation.

Ed Foote has noted these phenonomae independently and I have entered this thread only because what I have been doing corroborates all that he has been saying.

I encourage you simply to try tuning a piano in a good temperament (UT) and to experience the effects, or tune one for a musicologist to note its interaction with repertoire. If UT tunings can have the benefit of Alfredo's mathematical analysis of concepts of harmonic accordance beyond my empirical practice, that has to vary between piano and piano, all the better . . .

In practice the central octave must be pure temperament, the treble octave above similarly certainly up to G5 and ideally most of the tenor octave, the octave below the middle certainly down to G3 or F3 or lower. I find that inharmonicity below G3 down to below the break on many pianos can start to vary wildly and find this really confusing in creating a reliably predictable result. From memory, I think that I referred to this on my last post in Jake's thread about my videos and discussion about how one copes with these caprices of the strings and weaves them into a rational scheme would be worthy of discussion and helpfully so. I gave inharmonicity figures for a clutch of instruments that I have worked on recently.

Best wishes

David P

Postscript:

I suspect that the piano tuner of my childhood talked of making some keys better than others and certainly my music teacher talked of keys having different characters. I thought I was a bad musician because I could not hear them . . . until I tuned UT . . . I suspect that achieving purely mechanical equality of differences between notes has really only come in with Communism and took hold in homes beyond the ivory towers of concert halls during the period of the Cold War. But with the fall of the iron curtain, looking beyond the confines of communism between the keys is a little overdue.

Edited by Unequally tempered (11/04/1106:19 AM)

_________________________
_______________________________David Pinnegar, B.Sc., A.R.C.S.- East Grinstead, Sussex, UK - - http://www.organmatters.com - _______________________________Restoring life to music . . . and music to life . . . and a good deal more!

I was expecting some simple answers, but I can only read words to which no meaning can be assigned.

..."I'm tuning modern Steinways as well as older instruments and on one the owner's mother comments (dispassionately and as a lay listener familiar with the instrument's standard sound) that she's never heard it so "in tune"."...

With all respect, I don't know what to do with that feedback.

..."The tweaking of temperament in the remote keys is not actually a great deal away from ET but the result is most wonderful purity elsewhere."...

Purity elsewhere? Where? In complex chords?

..."When harmonic accordance is achieved, to which Alfredo is striving, then the sound in the badly tuned keys becomes more pure but is musically interesting."...

I can not follow you: When is "harmonic accordance" achieved? "...badly tuned...pure...musically interesting"? Honestly, these words puzzle me. I'm still waiting for Ed Foote to deepen on his "harmonic" concept and his own ET tuning, and for you to explain what "real" music is about. Do think in terms of "accordance", and you may want to grasp what has recently been achieved. At some stage, you'll see that intervals and beats have all been harmonized; you will understand that intervals wider than an octave have found their "beat" rule, that all intervals have their assigned tensions, and the decanted color palette is wider than ever, if referred to intervals beat speed variety and combinations.

..."Furthermore, now with considerable experience of the repertoire and its relationship with UT, there are many composers of the 19th century who drove phrasing and musical construction through passages of turmoil in order to resolve them by way of stillness and purity. The chords chosen, even in poorly tuned UT keys chosen for "special effect" avoided notes of poor intonation. We have noted passages where Liszt quite litterally wanted to achieve the effect of skating lightly as if upon glass or ice, particularly to find solid land on the other side of the phrase or passage. Ubiquitous ET has blinded us to these effects."...

I've tried, but it seems impossible (for me) to understand how you can talk about "ubiquitous ET". Which ET have you experienced? Which ET could you tune? Was it the tuning of the temperament octave?

..."It may well be that the _beauty_ achieved by UT recordings does not come through in recording and standard electronic reproduction, particularly through the poor medium of computer compressed audio. But in real life, in the room, the effect can be particularly moving and, close to the piano, where effects are heard which cannot be recorded"...

It may well be that temperamental theory evolved because of those "special effects", that many musicians, close to the piano, could hear. I did ask why temperaments evolved, but you don't seem to be interested in history and evolution.

..."So in real life, rather than the mere abstraction of recorded sound particularly as by YouTube http://www.youtube.com/watch?v=nPvHq8HvTKg the performer is particularly aware, and possibly even more so than the audience, of the particular effects and this influences phrasing and accentuation."...

Let me say, in real life each piano and every tuner's tuning will produce different effects; it is the mere comparing of theoretical tunings that opens to mere abstraction. If you refer to real life you could have no problem: nobody could ever put our first ET into practice. If this is true, your abstractions have become an obsession.

..."I encourage you simply to try tuning a piano in a good temperament (UT) and to experience the effects, or tune one for a musicologist to note its interaction with repertoire. If UT tunings can have the benefit of Alfredo's mathematical analysis of concepts of harmonic accordance beyond my empirical practice, that has to vary between piano and piano, all the better . . .

David, as a matter of fact for centuries temperamental models have "sliced" frequency values, elaborating only a "foreground". Even 12 root of two ET did so, and payed dearly for some erroneous axioms. I departed wrong axioms and worked on "differences", that I see as the "background", in the idea that a perfect sound whole could be experienced in real life and represented numerically. Today, what theories of the past had not accomplished has been achieved. Now I can only suggest tuning practice and attentive work on the expansion of the temperament octave, this is how I shape my tuning into Chas form.

The start of unravelling what I'm saying perhaps is to identify to which "ET" I am referring - the answer to that is any tuning whereby the proportional relationship of the middle note to the ones at the sides of one triad is the same as the triad of the semitone up, the two triads having the same aural shape and effect.

Whatever ET has been used, it's been ubiquitous for the past century.

"Purity elsewhere? Where? In complex chords?"

In a sensible UT, purity robbed from the remote keys is given to the home keys with few sharps and flats.

Ed Foote will understand what I have been writing about.

Best wishes

David P

PostscriptSome more definitions:

"Rooted" and "unrooted" chordsRooted chords are those where the beat frequencies between two notes equate exactly or nearly exactly with a fundamental note of which the two notes sounding are on a harmonic series. If loud enough, the beat frequency is audible as a note - violinists know them as Tartini notes, I believe, and the effect is used on Acoustic Bass 32 organ stops.

Unrooted chords are those where beat frequencies of notes have no musical relation to either note, and are ignored.

Rooted chords, if the fundamental note is played or other notes also on the harmonic series "lock" together and are solid.

Unrooted chords have "no meaning" and are disconcerting.

Equal temperament is unpleasant as the beat note of a major third is 1/4 tone sharp.

"Harmonic accordance" When I refer to a piano in harmonic accordance I'm referring to notes falling near exactly upon the harmonic series of bass notes. In the Unequal temperament I use this occurs mainly in keys B flat, F, C and G with chords involving thirds and in a number of the remote keys with fifths. This gives a different effect to the keys making the home keys resonate warmly and the remote keys resonate stridently - but most of them resonate. The instrument "locks" together in different ways depending upon what key is employed.

_________________________
_______________________________David Pinnegar, B.Sc., A.R.C.S.- East Grinstead, Sussex, UK - - http://www.organmatters.com - _______________________________Restoring life to music . . . and music to life . . . and a good deal more!

David wrote:..."...to which "ET" I am referring - the answer to that is any tuning whereby the proportional relationship of the middle note to the ones at the sides of one triad is the same as the triad of the semitone up, the two triads having the same aural shape and effect."...

Perhaps I understand what you do not like: when the shape and effect of two chromatic triads don't sound different enough for you to hear a difference.

It is evident that you are referring to the ET tuning of the temperament octave only. In fact, we all know that the "expansion" of the temperament octave has never been strictly ruled. In other words, the expansion of the temperament octave is normally left to the tuner's own personal discretion.

So, what you, Bill, Ed are really referring to it is the tuning of 12 notes, i.e. when 12 notes are tuned to a perfect ET. And you argue that the perfect ET tuning of 12 notes is responsible for the loss of key character. You would say that, also listening to the entire piano, you do not get the chords hierarchy nor the emotions and atmospheres that you, me, us, all tuners, musicians and audiences should and could experience, were the temperament octave tuned to a WT. And that the ET tuning of 12 notes would betray and misinterpret the composer's intensions. Last, you say that ET, no matter how the individual tuner expands the temperament octave, is ubiquitous.

Greetings, As this thread unravels into fractal opinions, it may be of some utility to step back and get a global view of what we are talking about before we split nits and scrutinize esoteric, arcane, opinions. As I understand it, Alfredo holds the point that all 88 notes on a piano should be united in one whole, mathematically symmetrical unit. With this approach, there is no difference in the amount of tempering of the intervals. Bill Garlick basically taught us that proper tuning was an 88 note temperament, though you cannot hear thirds as anything but dissonant below F2,(critical band becomes inclusive). The original width of the temperament octave was thus in control of the stretch throughout, the clarity of the octaves, the purity of the fifths, and the overall level of stimulation that the tuning can cause. My point is that there is a physiological effect that humans exhibit when exposed to different levels of dissonance, and that this differentiation has musical value when a composer aligns his composition to take advantage of the effects of consonance and dissonance. These are the "harmonic resources" I spoke of , and I don't know of a clearer way to explain them than to say that the resources are differing levels of stimulation caused by differing levels of dissonance. It is scientifically proven we involuntarily respond to higher levels of dissonance with higher levels of attention, as measured by heartrate, blood pressure, galvanic skin response, pupil dilation and several other markers I can't recall. That composers would use strident keys for expressive-stimulative purposes and calm consonant keys for expressive-sedating reasons seems to me to be so logical that I haven't yet heard a more plausible reason for the key choices that were made between 1700 and 1900.

AC writes: >You say that ET tuning, in Bach's days, was not "normal". Please, would you be able to tell what is today's "normal" ET tuning? .<<

Today's "normal" ET is a tuning that evenly divides the comma between all 12 notes. Why are we even discussing what ET is? No particular approach changes the fact that every third is equally dissonant, thereby obviating any variety of physiological effect.

AC: I give fundamental relevance to the entire amount of notes, of intervals and chords ready to be arranged (in our case) on the keyboard. All intervals, inside and outside the temperament octave, can draw precise beat curves; we can weave all beat curves into a unique form, namely a sound whole.<<

Ok, that means that any difference between keys is no more than a difference in pitch. One who doesn't recognize pitchs as identities,("perfect pitch", which I do not have), will hear all keys as having the same level of stimulation.

I wrote>.. I am referring to the equality. If all your like intervals are not tuned exactly alike, it is not an equal temperament. If they are all tempered alike, there is no difference in the sound of like thirds, (unless you believe there is some magic that causes different keys to have different emotional qualities."<<

AC >, no magic but sensitivity. I do not think we need to theorize heavily tempered intervals anymore for the sake of contrast. Different keys do keep their different qualities on the basis of different levels of tension, established time after time by the fundamental tone and resonating within the entire sounding body.

Sensitivity to what? What are the "different levels of tension" that arise from the "fundamental tone". If you tune a piano in Chas tuning, but do it 100 cents flat from A-440, will the key of C maintain its level of tension, even though you are playing it a 1/2 up on the keyboard? I think not.

AC: In 12 root of two ET, in order to find something "equal" we have to translate the scale values in cents, but does that mean that "like intervals are tuned exactly alike"? Is that how you understand ET? For me, that is not even simplistic but a distorted representation, since we (you included?) temper ET thirds (and not only thirds) so that their beat rate speed can be progressive. What is "alike" then, in your view? <<

What is "alike" is the physiological effect. We hear logrhythmically, so the the level of stimulation is the same, regardless of the octave it is found in.

AC: Perhaps you call "clarity" what I would call cacophony. At the end, you like pure thirds, you can explain 21 cent thirds, but you hate ET 13. something thirds because they sound alike. Mhhhh..?

The 13 cent third is an important size, it equates to the tempering in the key of A and Eb in most WT's. I don't hate any size thirds. I am simply disgusted by the loss of resolutions that having everything the same produces. I hate hearing music stripped of important effects for the convenience of tuning.

When I wrote;.."I think removing the haze of tempering that hangs over the equal temperament allows the true harmonic colors to be displayed."...

AC writes>Here we are again onto the "color" conjecture, plus "harmonic", plus "true". I know that you refer "color" to your "pain and pleasure" experience, and how some keys should sound better than others.<<

You have misunderstood. I haven't said any key sounds better or worse. I have said that keys should have different characters, since they are used for different purposes in classical composition. The only pain I experience is that of sheer boredom listening to music modulate into more of the same. Color is a metaphor, "true harmonic colors" refers to the distinct sounds produced by intervals of different sizes. In an ET, of any persuasion you care to create, there is not physical difference between the keys, and we are left to sense the effects of modulation from an intellectual perspective, only. In a WT, that same modulation carries more information, and much of that information is sensual, not intellectual. The sensual difference is where the WT has the advantage, and for me, the enjoyment of music is the sensual experience. When I am listening, I am not trying to think, I am trying to feel.

AC> > But doesn't "harmonic" (from harmonia, â€śjoint, union, agreement, concord of soundsâ€ť) recall the USA motto "E pluribus unum", "Out of many, one"? And if "harmonic" refers to "one", if it refers to a "whole", which UT or WT, out of dozens, displays "true harmonic alternation of pain and pleasure"? <<

In the search for universal truth, we are always tempted to simplify. In music, harmony is not referring to "one", but rather, to the result of blending dissimilar pitches. Tempering creates a vibrato. Would it be boring is a singer was always using the same speed of vibrato? I think so. A skillfull vocalist will match differing levels of vibrato to the intent of the music. Horn players use various pitches for the same note, depending on the score. I submit composers used various keys for various intentions. Reducing the keys to the same tempering robs the music of effect. But not for all listeners, just the ones that have awakened to the additional complexity and richness that the WT palette offers. I am not tuning WTs because I like them, I am tuning them because paying customers love them. I also tune ET for the same reason. I submit there are two kinds of tuners, those that tune multiple temperaments and those that tune only one. Or, to phrase it another way, there are tuners that believe all temperaments have value, depending on venue, and there are tuners that believe they have found the one temperament that is superior for all music and all others are inferior. I suggest that the latter is the more limited expertise. Regards, They that have ears, let them hear.... Regards, It would be quite boring to always hear pure intervals, or always hearing 13.7 cent intervals, or always hearing 40 cent intervals. There are many WT's, but they all share the same organization; more accidentals in the key, more tempering in the tonic thirds. It is a simple palette, and a very logical one that composers of the era could work with.

Greetings, I wrote, (poorly), "We hear logrhythmically, so the the level of stimulation is the same, regardless of the octave it is found in."

"Logarithmically" was the word, by which I mean that the phase difference we are calling beats, color, character, etc., is sensed not on a linear scale, but an exponential one. The 13 cents in the F3-A3 3rd is not sensed as any more or less stimulative than 13 cents in the A3-C#4.

Thirds control the harmonic color of the triad. There, I said it, so, if we need to debate that, please, somebody, anybody, call up another thread name!

The raising of pitch is often associated with increasing tension, and in ET, a move upwards is likely to have that effect. If, however, you are playing the F3 triad with a 8 cent third and modulate to an A triad with a 15 cent third, you are going to react involuntarily with a slightly higher degree of anticipation, (as in,"Where the heck is that Bb I am needing to hear!!?). regards,

..."As I understand it, Alfredo holds the point that all 88 notes on a piano should be united in one whole, mathematically symmetrical unit. With this approach, there is no difference in the amount of tempering of the intervals."...

ED, it seems to early to talk about Chas ET, maths and beat symmetries, let's talk about ET in general. I needed to understand how you interpret ET (in general), and for this reason I wrote: In 12 root of two ET, in order to find something "equal" we have to translate the scale values in cents, but does that mean that "like intervals are tuned exactly alike"? Is that how you understand ET? For me, that is not even simplistic but a distorted representation, since we (you included?) temper ET thirds (and not only thirds) so that their beat rate speed can be progressive. What is "alike" then, in your view?

Today you reply:..."What is "alike" is the physiological effect. We hear logarithmically, so the the level of stimulation is the same, regardless of the octave it is found in."...

So you too say that ET thirds beat rate speed is progressive, but (you say) "â€¦We hear logarithmically, so the level of stimulation is the same".

I now understand that you talk about "stimulations" and dissonances that go beyond the idea of "proportion". We do hear logarithmically, perhaps we are able to aurally detect a sense of proportion and this, in my view, is how we can tell whether instruments and singers are "in tune" or not. Now I understand that your argument it is not really "variety in beat rate speed", but dissonances that in WTs mess up that sense of proportion. You aim at consonant and over-dissonant intervals, so that we can get in and out that sense of proportion, so our emotions can flow. My emotions don't flow in the same way as yours and, in my opinion, once we get out of proportions, there is very little left that we can share.

..."Bill Garlick basically taught us that proper tuning was an 88 note temperament, though you cannot hear thirds as anything but dissonant below F2,(critical band becomes inclusive). The original width of the temperament octave was thus in control of the stretch throughout, the clarity of the octaves, the purity of the fifths, and the overall level of stimulation that the tuning can cause."...

How do you tune ET 4ths and 5ths, octaves, 12ths and 15ths? How do you expand your ET temperament octave? Did Bill Garlick tell you what the rules (outside the temperament octave) were, in order to achieve a "normal" 88-notes ET form? Did he tell you that we needed strict rules for the whole piano?

..."My point is that there is a physiological effect that humans exhibit when exposed to different levels of dissonance, and that this differentiation has musical value when a composer aligns his composition to take advantage of the effects of consonance and dissonance. These are the "harmonic resources" I spoke of , and I don't know of a clearer way to explain them than to say that the resources are differing levels of stimulation caused by differing levels of dissonance. It is scientifically proven we involuntarily respond to higher levels of dissonance with higher levels of attention, as measured by heartrate, blood pressure, galvanic skin response, pupil dilation and several other markers I can't recall. That composers would use strident keys for expressive-stimulative purposes and calm consonant keys for expressive-sedating reasons seems to me to be so logical that I haven't yet heard a more plausible reason for the key choices that were made between 1700 and 1900."...

You say "...logical" and "plausible reasons..."; I remember offering other plausible reasons but, in any case, "That composers would use strident keys for expressive-stimulative purposes and calm consonant keys for expressive-sedating reasons..." needs to be proved. You say: they had WT's; WT's had different key characters; thus composers exploited WT's for stimulative/sedating reasons. It becomes a problem when, against evolution and musical analysis, we need to prove the validity and the exclusivity of your sequenced points.

AC writes: >You say that ET tuning, in Bach's days, was not "normal". Please, would you be able to tell what is today's "normal" ET tuning? .<<

..."Today's "normal" ET is a tuning that evenly divides the comma between all 12 notes. Why are we even discussing what ET is? No particular approach changes the fact that every third is equally dissonant, thereby obviating any variety of physiological effect."...

Lucky we can discuss what ET is. I was not asking about theory but tuning practice. As I have mentioned, ET is not the tempering of 12 notes only, but a precise geometry that must be extended to the whole keyboard; in order to expand the ET temperament octave, what you call stretching, tuners do not follow precise rules; consequently, each individual tuner (or ETD) tunes a variant of 12 root of two; sum up these elements and be sure that "normal ET" (the normal ET you would not discuss) does not exist.

AC: I give fundamental relevance to the entire amount of notes, of intervals and chords ready to be arranged (in our case) on the keyboard. All intervals, inside and outside the temperament octave, can draw precise beat curves; we can weave all beat curves into a unique form, namely a sound whole.<<

..."Ok, that means that any difference between keys is no more than a difference in pitch. One who doesn't recognize pitchs as identities, ("perfect pitch", which I do not have), will hear all keys as having the same level of stimulation."...

I'm afraid we are going in circles. That means that all intervals, also those wider than the octave, have their precise proportional tensions. This, if anything, is how we achieve variety of stimulations. And I'm sure, if you were to be "stimulated" you would like to think that all tensions, all dissonances are under control. Once again, for this to occur you need to rule the expansion of your temperament octave.

ED wrote>.. I am referring to the equality. If all your like intervals are not tuned exactly alike, it is not an equal temperament. If they are all tempered alike, there is no difference in the sound of like thirds, (unless you believe there is some magic that causes different keys to have different emotional qualities."<<

AC >, no magic but sensitivity. I do not think we need to theorize heavily tempered intervals anymore for the sake of contrast. Different keys do keep their different qualities on the basis of different levels of tension, established time after time by the fundamental tone and resonating within the entire sounding body.

..."Sensitivity to what? What are the "different levels of tension" that arise from the "fundamental tone". If you tune a piano in Chas tuning, but do it 100 cents flat from A-440, will the key of C maintain its level of tension, even though you are playing it a 1/2 up on the keyboard? I think not."...

I'm not sure I understand your question, in any case you can check (*): tune your version of ET, play Bb2 together with Bb4-D5-F5, and transpose all the equivalent to F2 and E3, perhaps you too will hear different levels of tensions.

AC: In 12 root of two ET, in order to find something "equal" we have to translate the scale values in cents, but does that mean that "like intervals are tuned exactly alike"? Is that how you understand ET? For me, that is not even simplistic but a distorted representation, since we (you included?) temper ET thirds (and not only thirds) so that their beat rate speed can be progressive. What is "alike" then, in your view? <<

..."What is "alike" is the physiological effect. We hear logarithmically, so the the level of stimulation is the same, regardless of the octave it is found in."...

You have related "levels of stimulation" to dissonances and beats variety. Now I know you'd like over-dissonances, yet I would never say that ET thirds sound alike.

AC: Perhaps you call "clarity" what I would call cacophony. At the end, you like pure thirds, you can explain 21 cent thirds, but you hate ET 13. something thirds because they sound alike. Mhhhh..?

..."The 13 cent third is an important size, it equates to the tempering in the key of A and Eb in most WT's. I don't hate any size thirds. I am simply disgusted by the loss of resolutions that having everything the same produces. I hate hearing music stripped of important effects for the convenience of tuning."...

Convenience of tuning? We are not discussing that, but tuning models and practice through history and at present times.

Ed:..."I think removing the haze of tempering that hangs over the equal temperament allows the true harmonic colors to be displayed."...

AC writes>Here we are again onto the "color" conjecture, plus "harmonic", plus "true". I know that you refer "color" to your "pain and pleasure" experience, and how some keys should sound better than others.<<

Ed..."You have misunderstood. I haven't said any key sounds better or worse. I have said that keys should have different characters, since they are used for different purposes in classical composition."...

It sounds like you are turning your conjecture into an axiom. This kind of "expansion" is a problem.

..."The only pain I experience is that of sheer boredom listening to music modulate into more of the same."...

Yes, I understand your pain. It happens to me too, when out of tune intervals modulates into more of the same.

..."Color is a metaphor, "true harmonic colors" refers to the distinct sounds produced by intervals of different sizes. In an ET, of any persuasion you care to create, there is not physical difference between the keys, and we are left to sense the effects of modulation from an intellectual perspective, only."...

What is more intellectual than the abstract representation of an ET model that nobody could put into practice. I think you are not considering the gap between the first ET and our tuning practice, then you may suffer from the effects of your own perspective.

..."In a WT, that same modulation carries more information, and much of that information is sensual, not intellectual. The sensual difference is where the WT has the advantage, and for me, the enjoyment of music is the sensual experience. When I am listening, I am not trying to think, I am trying to feel."...

For me that information might not be sensual at all. You translate WT informations and your sensual experience into WT vs ET differences, this is intellectual. In regard to music, I'm sure everybody would go for feelings.

AC> > But doesn't "harmonic" (from harmonia, â€śjoint, union, agreement, concord of soundsâ€ť) recall the USA motto "E pluribus unum", "Out of many, one"? And if "harmonic" refers to "one", if it refers to a "whole", which UT or WT, out of dozens, displays "true harmonic alternation of pain and pleasure"? <<

..."In the search for universal truth, we are always tempted to simplify. In music, harmony is not referring to "one", but rather, to the result of blending dissimilar pitches."...

We agree then, "the result of blending..." as one.

..."Tempering creates a vibrato. Would it be boring is a singer was always using the same speed of vibrato? I think so. A skillfull vocalist will match differing levels of vibrato to the intent of the music."...

Again, we agree!

..."Horn players use various pitches for the same note, depending on the score. I submit composers used various keys for various intentions. Reducing the keys to the same tempering robs the music of effect. But not for all listeners, just the ones that have awakened to the additional complexity and richness that the WT palette offers."...

Try the experiment above (*), then tell me whether the vibratos are the same or what.

..."I am not tuning WTs because I like them, I am tuning them because paying customers love them. I also tune ET for the same reason."...

Really? Well, that is not my case. Leave my customers aside, I love Chas and the preparatory tuning.

..."I submit there are two kinds of tuners, those that tune multiple temperaments and those that tune only one. Or, to phrase it another way, there are tuners that believe all temperaments have value, depending on venue, and there are tuners that believe they have found the one temperament that is superior for all music and all others are inferior. I suggest that the latter is the more limited expertise."...

I guess it's my 'expert' tuning ear that I have acquired since learning how to tune....lol....there are a few treble unisons out...its a bit disconcerting every time they are struck...never the less, the piece is very pleasant to listen to...and I enjoyed their interpretation.

You are welcome. Thank you for your comment, I too find that recording very enjoyable, that's one of the reasons why I decided to share it despite some tuning and voicing imperfections. One month later this young pianist, Federico Colli, won the 1st prize at the 2011 Mozart International Piano Competition, in Salzburg.

Another reason is that it is not so frequent to be able to release a recording in this way, and I think it is very generous on Federico Colli's part getting involved in the divulgation of Chas ET.

[Re: Weiyan] rxd: ..."Our wide M3rds, for example, are regarded as a problem inherent to in equal temperament. It is anamolous that players of melodic instruments and singers hear them even wider. Its inversion, the minor 6th they want to play or sing very wide (melodically) and is more of a 'problem' with ET. This is where unequal temperaments really gain favour with singers."...

Hi rXd,

May I ask you to expand on the above statements? Ordered one by one:

1 - Our wide M3rds, for example, are regarded as a problem inherent to in equal temperament.2 - It is anamolous that players of melodic instruments and singers hear them even wider.3 - Its inversion, the minor 6th they want to play or sing very wide (melodically)...4 - ...and is more of a 'problem' with ET.5 - This is where unequal temperaments really gain favour with singers.

Sorry. I don't normally follow temperament threads and I was only just told this was here. The intervals of vertically structured temperaments and the melodic intervals as perceived and performed by say, gypsy violinists and opera singers simply cannot be reconciled.

Shifting pitch bases is the answer to this quandary. I am currently researching the BBC archives for a recording that exemplifies this extremely well.

I remember being called in to a piano trio recording where the musicians could not understand why the 'cellist had a few measures solo and his final note did not match the piano entry. His intonation in the solo was exemplary string players intonation. So much so that he gained pitch in the few measures that he played alone. I advised him not to play a minor 6th not quite so wide and a he finished in tune with the piano. It took a great deal of concentration on the 'cellists part but he was not "wrong". The result is out there on a commercial recording somewhere. I didn't think enough of it to remember the label never thinkng that this information might one day become useful.

A little thought would make it evident that, even if I had re-tuned an unequal temperament on the piano, this problem would still exist, perhaps in a more exagerated form.

This has little to do with the title of this thread but I am willing to expand on this should anybody be interested.

Yes, I'm interested in knowing your thoughts, hearing about your own tuning experience and analyzing some conclusions, and I hope some other colleagues too are willing to share their own experience and deepen on one crucial issue, namely "intonation".

You said: â€¦"The intervals of vertically structured temperaments and the melodic intervals as perceived and performed by say, gypsy violinists and opera singers simply cannot be reconciled."â€¦

That seems to recall two kinds of "intonation", one for vertical chords structures and a second one for horizontal melodies, based on how intervals are perceived. Is that what you are referring to?

You kindly reported one case and said: â€¦"His intonation in the solo was exemplary string players intonation."â€¦

Do you have other cases? What would you say is "string players intonation" like?

And, if you would like, we could cover some other questions from your other post (above), at your convenience:

Who does regard our wide M3rds as a problem? Do you?Would you yourself hear ET wide M3rds even wider?How wide would players of melodic instruments and singers hear minor 6ths?How do you expand your equal tempered octave?Where do unequal temperaments gain favour with singers? Do you refer that to UT 6th? To one precise key?

In all this sharing, I would like to make one point and three questions.

In my opinion, 12th root of two ET may well be considered a Historical Temperament too, since it pays for the "pure octaves" ancient dogma, and today it could well be referred to as the first algebraic/geometrical model, just by acknowledging other new ET models.

Why do I get the impression that Time stopped with 12th root of two ET?

People featuring non-equal temperaments or UTs say that they have more "colour", that "tempering" from just results in (in meaning) tone's colour. This may be fair enough. Personally, I'm in favour of beats and you may well know why.

What I do not understand is: what is difficult about acknowledging modern ETs, i.e. new algebraic geometrical models, new degrees of harmoniousness, new tonal effects, new spectral fusions, and accepting that also 12th root of two ET could evolve, actually it has evolved.

What is then "true" ET?Does (in your opinion) modernity make people giddy?

a.c.

.

I know that this thread was started a few years ago, but I see that it is still active. I'm not a piano technician or tuner, but I am a graduate student in mathematics who studies number theory and a bit of music theory. I saw a few posts talking about / questioning the validity or long-time-use of the 12-TET tuning system. If you anybody is willing to deal with some basic high school mathematics and is interested in the subject, I can try to illuminate the construction of 12-TET to show where it comes from and at the same time explain its strengths and weaknesses.

It is very kind of you offering your knowledge and mathematical tools in order to expand on our (can I say?) historical ET and, as you say, on "the construction of 12-TET". I'm sure that like myself, many tuners and musicians will appreciate that, if not all.

You also mention "...where it comes from...", "...its strengths and weaknesses.", I look forward to reading about that too.

Originally Posted By: AlfredoI'd rather talk about the "lack introduced by" having to manage the wolf, both in theory and practice. And whether you sing in a choir or play in an orchestra, or brass or wind band... you'd still have to manage complex chords, correct?

Tunewerk wrote: ..."Actually, you just rephrased exactly what I said. The wolf results from the keyboard compromise.

In a choir, you do not have to manage this. This is a critically important technicality.

Voices auto-adjust to find pure intervals or complex chords adjusted within themselves for optimum consonance. It is the fixed scale that forces the wolf compromise.

A choir is the same thing as an auto-adjusting keyboard would be: each string moving slightly to accommodate the new chord for the desired effect or consonance."...

- . - . - . -

I apologize, I realize that I was jumping from one issue to another one, so perhaps I should first re-state my point: There is no way we can sing three (or more) notes simultaneously, and obtain pure (perfectly consonant) intervals only. In other words, when more than two notes are played simultaneously, we end up having to face the same (fixed scale) problem: how do we (referring to any ensemble) make complex chords sound "in tune"?

Originally Posted By: AlfredoWhat about violin players or singers, when they practice... do they (think about the 9 commas between whole tones)?

Tunewerk wrote: ..."Yes, they do. But they do not typically know it. The advanced players do know this and understand. They auto-adjust to 9 commas or more to get correct intonation on specific intervals and scales. This is what makes a great violinist."...

Well, I have a different opinion, perhaps based on my personal observation and experience: singers and all bows, actually all ear equipped musicians, practice in order to get correct intonation not on specific intervals, as if they were to prefer pure intervals, they practice intonation and scales within the whole tonal complexity. But I agree, they do not typically need to know, and yes, great musicians are capable of "bending" notes consciously (on lyrical (slow) passages only), depending on the desired effect or perhaps in the effort to improve the ensemble overall intonation. Is this what you mean, that fixed tone instruments cannot improve the overall intonation in real time? I would agree.

But in general, in my view, when going for "correct" intonation, we do not refer to pure consonance anymore (as that is simply not possible, unless we are willing to crash other intervals), but to the best distribution (allocation?) of all commas, that is also what we (would) do on fixed tones instruments.

I know I'm going against a well established idea, that non-fixed tone instruments can "adjust" or as you say "auto-adjust", and so achieve better/correct intonation in absolute terms but, if this is what you think, I ask: isn't all this related to the questionable idea that intervals, at least some intervals, in order to sound "in tune" should be pure? Isn't it true that pure consonant intervals would cause other intervals be less consonant? In case more than two notes are played simultaneously, what would they try to adjust on?

I just stumbled upon this. I'm honored that you would consider reposting my information, and of course you are welcome to.

I am posting here because I want to clarify and not be misunderstood. It's difficult to talk about abstract concepts on this message board where one cannot interject to clarify..

Originally Posted By: Alfredo

There is no way we can sing three (or more) notes simultaneously, and obtain pure (perfectly consonant) intervals only. In other words, when more than two notes are played simultaneously, we end up having to face the same (fixed scale) problem: how do we (referring to any ensemble) make complex chords sound "in tune"?

This is possible. It just requires movable, unequal divisions in-between. Let's run a little mathematical proof:

Let's say we have a choir at our disposal and order a perfect open fifth to be sung:

Tonic Voice = 1Harmony #1 = 3/2

Here we have two notes and one interval interaction [2,1].

Now, we add a third:

Tonic Voice = 1Harmony #1 = 3/2Harmony #2 = 5/4

Now we have three notes and three interval interactions [3,3].

Let's look at it. We have the tonic note, at 1, the perfect M3rd at F1*[5/4] and the perfect 5th at F1*[3/2]. That's two relationships, so what about the third? This interval is described by [3/2]*[4/5], which equals exactly [6/5], the perfect m3rd.

The reverse justification, which might be easier to think about is [5/4]*[6/5] = [3/2].

Extending outward for all chordal combinations, this runs a geometric progression of [(n^2-n)/2]:

[1,0][2,1][3,3][4,6][5,10]etc..

So, let's look at another chord, this time with 4 notes:

Tonic Voice = 1Harmony #1 = 3/2Harmony #2 = 5/4Harmony #3 = 9/5

This is more interesting, adding on the most common pure ratio approximation for the just dominant 7th. Point 4 from our series yields 6 interactions, so we should list them out:

To get deeper into other chordal combinations yields the same result: non-irrational pure interval consonances. I have not completed a mathematical proof out to infinity, but these relationships are well known in analytic number theory.

Any combination of rational consonant values, will have rational consonance in-between. The consonance may be thin, but it will be a traceable pure rational interval.

Another way to think about it is: Each time a vocal chord is formed, a new geometric shape is being formed mathematically. The consonance ratios build the sound outwards into a distinct shape that by law of its rational parts, has complete consonance through and within.

This is not to say there wouldn't be any beats.. depending upon the harmonics generated by the voices. Any closely coinciding partials will generate beats and one would have to calculate the intensity, bandwidth and location of all generated partials within relative proximity to see if a few intersected close enough to cause beat effects.

Standard equal temperament aligns all notes by the equal logarithmic division of the 2^x curve. This produces irrational frequency values (numbers not generated by an integer ratio), constrained to the 2^x curve. Combining these fixed tones in chord structures introduces beats into any chord, except for octave derivatives, because none of the other ratios align to the 2^x curve (only 2/1). Further misalignment is found as partial value is increased.

If the resolution of the 12-TET scale were increased, this would help the problem, the best solution being 53-TET under the 60-tone division threshold. However, finer division will never purely solve the problem, it will only introduce finer levels of hysteresis around the 2^x curve. I posted earlier about this. Here's an image of what's going on in standard cents:

If a flexible string instrument were used, the geometries could change for each harmonic and melodic alignment to provide perfect consonance continuously with new geometries internally aligned, not preformed to the 2^x curve. Each time the chords changed, the frequency values would shift.

I suppose this takes off directly into derivation of the 12-tone scale and its mathematical basis. Here's an illustration I previously posted as well, showing total deviation in relative cents (1/100 of the native semitone) for all ET octave division schemes up to 120-TET.

It's clear, from the deviations shown, for all note division schemes under 18, 12-TET is the best solution. I wonder how this was first found empirically?

These are all the inclusive major deviations for ET schemes up to 1000-TET. The lower the dot appears, the purer the yield that scale division provides. Good scales could be thought of as those that yield lower than 40 relative cents cumulative deviation for the 4 major intervals.

To get deeper into other chordal combinations yields the same result: non-irrational pure interval consonances. I have not completed a mathematical proof out to infinity, but these relationships are well known in analytic number theory.

You use terms here like "geometric progression" and "analytic number theory", but I'm pretty sure that isn't what you mean - neither of those terms apply here. But in any case, in the first part of this quote, it seems to me that you are trying to count the number of pairs of tones (i.e. dyads) you can make if you have N tones to work with. If this is the case, there is a result from combinatorics / graph theory called the Handshake Lemma which gives you the correct formula. The Handshake Lemma says that if there are N people who want to shake hands with one another, there will be a total of N(N-1)/2 (or equivalently, (N^2-N)/2) handshakes. This is exactly the formula that you were trying to justify (thinking of a dyad combination as a handshake).

In the second part of the quote, you say that multiplying combinations of rational numbers will be "non-irrational". Actually, this is true almost by definition of rational numbers and their arithmetic. To prove this, take two fractions a/b and c/d. Then, (a/b)*(c/d) = (ac)/(bd), which is rational. That this is true for any (finite) number of multiplications should be clear.

Quote:

I suppose this takes off directly into derivation of the 12-tone scale and its mathematical basis. Here's an illustration I previously posted as well, showing total deviation in relative cents (1/100 of the native semitone) for all ET octave division schemes up to 120-TET.

[A chart was shown here]

It's clear, from the deviations shown, for all note division schemes under 18, 12-TET is the best solution. I wonder how this was first found empirically?

There is actually a way to analytically obtain the result that 12-TET is the one of the best rational approximations for the exponential function that you referred to a few times. (You can do better if you are willing to accept more tones.) I hope to be able to say more about it when I get the time. The solution uses some basic arithmetic and logarithms, so I hope that it will be accessible.